Consider the spinc structure on the flat standard 3-torus, which you get from the trivial (or any other) spin structure. Its associated vector bundle can be identified with a trivial bundle with fibre $\mathbb{C}^2$. spinc Dirac operators on this bundle are parametrized by one-forms and look like $D_\alpha = D_0+ic_\alpha$, where $D_0$ is the spin Dirac operator and the $c$ means Clifford multiplication.

My aim is now to find a spectral decomposition for $D_\alpha$. If α is closed, this can be easily done by reducing everything to the case where α is harmonic. The case where α is not closed seems to be more tricky, so I would like to ask the community:

A suggestion, can you gauge transform so that $\alpha$ is nice? Gauge transformations tend to preserve spectral decompositions.
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PaulFeb 17 '10 at 1:36

1

If you decompose &alpha; as $d\beta + d^*\gamma + \delta$, where $\delta$ is harmonic, then you can "gauge away" the d\beta-Part by the usual $U(1)$-gauge. Since &delta; is constant, this gives you the solution for &alpha; closed. I don't know what kind of gauge should work on the $d^*\gamma$-part.
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J. Fabian MeierFeb 17 '10 at 10:52

To find ${\rm spec}(D_\alpha^2)$ you need to understand spectrum of ordinary differential operators of the form

$$ -\partial^2_\theta + A(\theta) $$

acting on functions $u: S^1 \to \mathbb{C}^2$ where $A(\theta)$ is a $2\times 2$ complex hermitian matrix depending smoothly on $\theta\in S^1$. I don't know how to find the spectrum of such an operator but maybe you can find something in the literature.